Module 7 Test Review

Now is a chance to review all of the great stuff you have been learning in Module 7! Solutions to Equations and Inequalities Applications with Expressions Solving Equations Inequalities Dependent and Independent Variables Analyzing Relationships

Key Terms Equation: mathematical sentence that shows two expressions are equal using the equal sign Solution: Any value substituted for a variable that makes the mathematical sentence true

Example of an equation and solution

Inequalities Inequality: A mathematical sentence that shows a comparison between two expressions using the less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥) symbols.

Inequalities The inequality symbols “is less than or equal to” (≤) and “is greater than or equal to” (≥) are like two symbols in one. Think of a statement that uses one of these symbols as a combination of an inequality and an equation. If either the inequality or the equation is true, then the entire statement is true.

Inequalities - Examples Is x ≤ 8 true, when x equals 5? The statement 5 ≤ 8 is true if either the statement 5 = 8 or the statement 5 < 8 is true. 5 = 8 is false. 5 < 8 is true. Because 5 < 8 is true, the inequality 5 ≤ 8 is true because 5 is less than 8. The solution can be less than or equal to as it cannot be both.

Sets of Numbers Because inequalities compare two expressions, there are multiple values that can make the statement true. Sometimes, you may have to check multiple values that are presented in a set.

Sets of Numbers - Example Which value or values from the set {1, 3, 5} make the inequality 4x + 8 > 12 a true statement? How do you know? Substitute each value from the set into the inequality to see which values make a true statement. Substitute 1 into the inequality and simplify 4x + 8 > 12 4(1) + 8 > 12 4 + 8 > 12 12 > 12 Is this a true statement? This is not a true statement. The value 12 is not greater than 12, so x = 1 is not a solution. Substitute 3 into the inequality and simplify 4(3) + 8 > 12 12 + 8 > 12 20 > 12 This is a true statement. The value 20 is greater than 12, so x = 3 is a solution. Substitute 5 into the inequality and simplify 4(5) + 8 > 12 20 + 8 > 12 28 > 12 This is a true statement. The value 28 is greater than 12, so x = 5 is a solution.

Writing Algebraic expressions Being able to write algebraic expressions that represent real-world situations is very important in evaluating problems Do you remember the meaning of the most common action phrases? Addition terms Added to More than Increased by Plus Sum Total Subtraction terms Minus Less than Subtracted from Difference between Decreased by Take away Fewer than Multiplication terms Doubled Product Twice Times Per Multiplied by Tripled Division terms Half Ratio Quotient Divided by Variable terms A number A variable Some number Unknown value Exponent terms Cubed Power Squared

Writing Algebraic expressions When translating a real-world problem into an algebraic expression, it helps to follow a few simple steps. Step 1: Identify the action phrases. What are the mathematical operations? Step 2: Define the variable. What is the unknown quantity in the situation? Step 3: Translate the sentence into a verbal expression. Step 4: Translate the verbal expression into an algebraic expression.

Example The Problem Jackson needs to purchase some shoes and socks. Shoes cost $3 more than twice the price of the socks. Write an algebraic expression to represent the cost of the shoes Step 1: Identify the action phrases. There are two mathematical phrases in this problem: "more than" and "twice." Therefore, you know this problem includes multiplication and then addition.

Example Step 2: Define the variable In this scenario, determining an expression for the cost of the shoes is the goal. To determine this, you must calculate twice the cost of socks. Because you do not know the cost of socks, the variable in this problem is the cost of the socks. You can represent this unknown number with the letter s.

Example Step 3: Translate the sentence into a verbal expression. "$3 more than twice the socks" becomes "3 more than twice a number." Step 4: Translate the verbal expression into an algebraic expression. The verbal expression "3 more than twice the socks" can be written as 2s + 3, where s represents the cost of socks. This means that Jackson can use the expression 2s + 3 to determine the cost for shoes, and the variable s represents the cost of socks.

Writing Equations You already know how to translate expressions, and translating real-world problems into equations uses a similar skill set. Just locate the words that indicate the equal sign. "some number plus 20 is 55" is an example of the verbal form of the equation x + 20 = 55. The verb "is" tells you where to place the equal sign. Other mathematical words that indicate an equal sign are "equals," "yields," and "results in." Be sure to add them to your list of common action phrases

Solutions to Equations You just solved an algebraic equation using one strategy; however, there are other strategies you can use.

Solving with Opposites You can also use inverse operations to solve an equation. Inverse Operation: An operation that reverses the effect of another operation; for example, adding 3 and subtracting 3 are inverse operations. When you have an operation and an inverse operation together, they will “undo” each other.

Golden Rule for Solving Equations What you do to one side of an Equation, you MUST do to the other side of the equation. To solve an equation you must isolate the variable on one side of the equal sign by using inverse operations. This results in zero by undoing the operation

Example Solve the equation x − 7 = 18 to determine the value of x. To determine the value of x, you must isolate it on one side of the equation. Use inverse operations to get the variable alone. Operations Explanation x − 7 = 18 x − 7 + 7 = 18 + 7 Use the inverse operation of subtraction by adding 7 to both sides of the equation and simplify. X = 25 The solution to the equation is 25.

Try it!

Check your work

Using the Inverse The inverse operation of multiplication is division. To "undo" multiplication in an equation you must divide the number on both sides of the equation to isolate the variable.

Using the Inverse - Example Use inverse operations to solve the equation 10k = 60 for k. The equation has k being multiplied by 10. To get k alone on the left side of the equal sign, you must use the inverse operation of multiplication. So, divide both sides of the equation by 10. Next, cancel the common factors in the numerator and denominator, then simplify the equation. Therefore, the solution to the equation 10k = 60 is k = 6.

Using the Inverse The inverse operation for division is multiplication. If an equation has a coefficient that is a fraction, you just multiply both sides of the equation by the reciprocal of that coefficient.

Using the Inverse - example

Using the Inverse - example

Inequalities Recall, inequalities are known to have a range of solutions. In fact, any number that satisfies the condition is a solution.

Example Write an inequality to represent the solution of the situation given. The spring fair committee must have more than 200 people attend the event. Define the variable: The variable in this situation is the number of people that must attend the event. Let x represent the unknown number of attendees. Rewrite the situation: Sometimes, the most challenging part of writing an inequality from a word problem is deciding which inequality symbol to use. If 200 people attend, will the committee reach its goal? No! The goal is more than 200, so 200 is not included in the solution. This means you must use the greater than symbol. The number of attendees must be greater than 200.

Let’s Draw – Using a Number line Graphing the values that make an inequality true requires the number line. But inequalities can have an unlimited number of solutions. Also, there are four types of inequalities. How do you graph all possible solutions for the different types of inequalities?

Graphing Inequalities - Example Greater Than Inequality: x > 3 Graph: Explanation: The inequality symbol is greater than. After drawing a number line, use an open circle on 3. Shade everything on the number line to the right to include all the numbers greater than 3. The graph shows that any number greater than 3 is a solution. However, 3 is not a solution.

Graphing Inequalities - Example Less Than Inequality: x < 3 Graph: Explanation: The inequality symbol is less than. After drawing a number line, use an open circle on 3. Shade everything on the number line to the left to include all the numbers less than 3. The graph shows that any number less than 3 is a solution. However, 3 is not a solution.

Graphing Inequalities - Example Greater Than or Equal to Inequality: x ≥ 3 Graph: Explanation: The inequality symbol is greater than or equal to. After drawing a number line, use a closed circle on 3. Shade everything on the number line to the right to include all the numbers greater than 3. The graph shows that 3 and any number greater than 3 is a solution.

Graphing Inequalities - Example Less Than or Equal to Inequality: x ≤ 3 Graph: Explanation: The inequality symbol is less than or equal to. After drawing a number line, use a closed circle on 3. Shade everything on the number line to the left to include all the numbers less than 3. The graph shows that 3 and any number less than 3 is a solution.

Key Terms Dependent variable: A variable with a value that is dependent on the value of another variable. Independent variable: A variable that affects another variable; it may have its value freely chosen without considering the other variable's value.

Using Independent and Dependent Variables Peter is another student at the science fair. He studied the effects of air resistance on paper airplanes. He did so by adding flaps to the wings of the paper airplane and watching how far the plane travels. Determine the independent variable and Dependent variable

Using Independent and Dependent Variables You can identify the independent variable by determining which variable is affecting the other. Peter is changing the amount of air resistance, so this is the independent variable. The dependent variable depends on changes to the independent variable. The flight distance changes depending on the amount of air resistance. Therefore, flight distance is the dependent variable.

Analyze the relationship Now that the information is recorded, give the table another look to see if you notice a pattern. If you look closely at the changes in circumference when the diameter is changed, you can see that for a 1-inch increase in diameter, these is a 3.14-inch increase in circumference. This is a special ratio that exists between the diameter of a circle and the circumference of the circle.

Writing Equations Relationships between two variables can be analyzed even further by writing equations Before you write equations to represent relationships, you must define the variable. In general, the variable x is used for the independent variable and the variable y is used for the dependent variable. Let’s Look at this further: In an electrical circuit, the current (measured in amperes) is related to the voltage supplied from batteries. Another student investigates this relationship in his science fair project. He is going to increase the number of batteries in a circuit and record the current in the circuit for each number of batteries.

Writing Equations Show the variables and the recorded information The number of batteries is the variable that is changing. The number of batteries is the independent variable. The current goes up or down depending on the number of batteries. The current is the dependent variable.

Write an equation that represents this relationship Define variables: Let B equal the number of batteries (the independent variable). Let C equal the amount of current (the dependent variable). Looking at the table, you can see that for one battery, the current was 0.8 ampere. Every one increase in the number of batteries causes the current to increase 0.8 ampere. The equation C = 0.8B represents this relationship.

Verify the equation It is very helpful to check the equation for accuracy, so let's do that this time. To check this equation, you can substitute values from the table. What is the value for C if B = 4? C = 0.8B C = 0.8 ⋅ (4) C = 3.2, which matches the value in the table. The equation and the table both show the same relationship.

Creating Graphs We can use the information we gather from a table or equation to create a graph. Let’s look at a scenario Every day for six days, Anthony measures the total amount of water that escapes from a leaky faucet. Here is his data table. He finds the leaky faucet wastes 0.5 liter of water each day. Based on this information, Anthony writes an equation to show his results. He lets x equal the number of days and y equal the amount of water wasted, in liters, and arrives at the equation y = 0.5x. Now he wants to make his science project look better by turning this information into a graph.

Creating Graphs Let's explore each section to see all of the information Anthony needs on his graph so that people can understand it clearly. The first thing Anthony needs is a title. A title tells a person looking at the graph a little bit about the variables being studied. It’s like a short introduction about the relationship being studied. An appropriate title for this graph is “Wasted Water.”

Creating Graphs Next, Anthony needs to label the x-axis and the y-axis to identify his variables. When graphs are created, the independent variable goes on the x-axis; so, he labels the x-axis “Number of Days.” The y-axis shows the dependent variable; so, he labels the y-axis “Amount of Water (liters).” The units of measurements are also shown to let people know the measurements that took place. Always remember to choose appropriate increments for the axes based on the values you graph.

Creating Graphs Title Labels

Creating Graphs Last, Anthony needs to plot on the graph the information he recorded. With each axis labeled, he can turn each row of the table into a set of coordinates. As mentioned, the x-coordinate is the independent variable and the y-coordinate is the dependent variable. Now the points are ready to be plotted.

Creating a Graph Plot the data

Creating a Graph In Anthony’s experiment, the data collected was continuous because the leaking water never stopped. Although Anthony took the measurements once a day, he really could have taken them at any time of the day. Because of this, he can connect his points with a line to show the reader that any point on the line is a possible value for his project. Now his graph is complete

Connect the points or Not to connect the points. Continuous: Numerical data that can take on any value within a range, such as decimals and fractions. Data that is continuous can have the data points connected on a graph. Discrete: Numerical data that can only take on certain values, such as only whole numbers. Discrete data DOES NOT have connected data points on a graph.

From Table to Equation Using the information in the following table, you can create an equation to show the relationship between the two variables. Let’s write an equation that shows the relationship between the number of months and the height of a plant.

From table to equation Identify variables The height depends on the number of months the plant has to grow. The height is the dependent variable. The independent variable is the number of months. As the number of months changes, it affects the height of the plant. Define variables Let y represent the height. Let x represent the number of months. Describe the relationship For each month of growth, the plant gains 2 inches in height. Write the equation In this scenario, the equation is y = 2x.

multiple representation Here are all four representations at once. Verbal For every hour of travel, a distance of 20 miles was traveled. Equation y = 20x Table Graph

Solutions to Equations and Inequalities Applications with Expressions You have now had a chance to review all of the great stuff you learned in Module 7! Solutions to Equations and Inequalities Applications with Expressions Solving Equations Inequalities Dependent and Independent Variables Analyzing Relationships Have you completed all assessments in module 7? Have you completed your Module 7 DBA? Now you are ready to move forward and complete your module 7 test. Please make sure you are ready to complete your test before you enter the test session.